Coordinate Geometry: JEE Main 2026 Mathematics question

Let $A$, $B$ be points on the two half-lines $x-\sqrt3|y|=\alpha$, $\alpha>0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $AB$ meets the angle bisector of the given half-lines at the point $Q$. If $PQ=\frac92$ and $R$ is the radius of the circumcircle of $\triangle PAB$, then $\frac{\alpha^2}{R}$ is equal to ______